One of the primary things the readings have reminded me of is how important it is to keep the connection between addition and subtraction within the forefront of whatever lesson we are doing. Addition is putting things together and subtraction is pulling things apart. We begin with a number and if we use the operation of addition we end up with a larger number. If we use the operation of subtraction on that number our answer will be smaller. Addition and subtraction both involve part-whole relationships. Addition is reconstructing a whole from parts; subtraction is reconstructing a part from the whole and the complementary part. If we subtract 5-2 together and get 3, it is only natural that we consider the reverse: 3+2 = 5.
In dealing with addition and subtraction without renaming, I would suggest that we are setting students up to work with larger numbers, and also helping students begin to understand the language of these two operations. How do we know when to add or subtract? If we give the example of
32
+ 43
------
we want the student to understand that he is adding the ones and the 10s, and arranging the numbers in vertical columns is a way to do this automatically. We would probably have available base ten blocks which the student could use. At this point they would see that all of their calculations go very easily across because they have a value of 9 or under in each column. Also to encourage algebraic number sentences the problem could also be written horizontally as 32 +43 =___. Either way we would hope that from their previous work in decomposing numbers students might easily see this as (30+2) +(40+3) =70 +5 = 75. Another possibility would be (30+40) + (2+3) =75.
Addition is straightforward and pretty easy to understand. Carpenter et. al. in
Children’s Mathematics
describes addition problem as joining together. Joe has 324 baseball cards. His mother gave him 15 more. How many baseball cards does Joe have now? In their book they continue exploring the problem types. They finally categorize eleven different problem types which I would prefer to simplify.
In
Guiding Children’s Learning of Mathematics
, Kennedy and Tipps simply state that all addition problems are basically joining of two or more groups. They split subtraction type problems into 4 categories. The first and most recognized is what most students would term “take- away” problems. I had 12 stickers. I gave 5 to my best friend. How Many stickers do I have left? Unfortunately by labeling subtraction as “take-away” students get a limited view of these types of problems. There are also comparison problems, completion problems and part/whole problems. Most of the literature advises not allowing children to think of all subtraction as only “take-away.”
In comparison problems we are comparing the size of two sets. For example: There were 345 people at the movie on Friday night. On Saturday night 289 people came. How many more people were at the Friday night show? Or we could ask: How many fewer people came on Saturday than came on Friday?
In the completion or unknown problem we are use subtraction to find a missing set which when put with a second set helps to make a third set. For example: The book Mary is reading has 123 pages. She has 97 read pages. How many pages does she have left? We subtract 123 -- 97 to find the 26 pages. Mary needs to complete the book..
My problem with the Carpenter book is that the authors describe the following as a joining problem: Tony had 5 toy trucks. His dad gave him some more for his birthday. He now has 9 trucks. Because the numbers in this problem are single digit the authors describe the children solving this with counting on from 5 to 9. However, if the numbers were higher --say, Tony starts with 47 trucks and that after his birthday he has 65 trucks- it is my opinion that this would not be a joining problem but a completion problem in which the students would simply subtract 63-47 = 16. Carpenter treats it as part of addition because in truth the action that happens in the problem ends with Tony having an increased number of toy trucks. To me it is still a subtraction problem..
